File size: 7,865 Bytes
f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.FreshCollar
import OAI.Probability.CriticalPercolation.Gluing.SourceGraph
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} [Countable V] [Countable E] (G : BondGraph V E)
omit [Countable V] [Countable E] in
lemma withSource_edge_cases (A : Set V) (e : E⊕A) (v w : Option V)
(he : (G.withSource A).ends e=s(v,w)) :
(∃ b x y, e=Sum.inl b ∧ v=some x ∧ w=some y ∧ G.ends b=s(x,y)) ∨
(∃ a : A, e=Sum.inr a ∧ ((v=none ∧ w=some a.val) ∨ (w=none ∧ v=some a.val))) := by
cases e with
| inl e =>
obtain ⟨⟨x,y⟩,hxy⟩ := Sym2.mk_surjective (G.ends e)
change s(x,y)=G.ends e at hxy
have hh : s(some x,some y)=s(v,w) := by simpa [withSource,← hxy] using he
rcases Sym2.eq_iff.mp hh with hh|hh
· exact Or.inl ⟨e,x,y,rfl,hh.1.symm,hh.2.symm,hxy.symm⟩
· exact Or.inl ⟨e,y,x,rfl,hh.2.symm,hh.1.symm,by simpa only [Sym2.eq_swap] using hxy.symm⟩
| inr a =>
right
refine ⟨a,rfl,?_⟩
have hh : s(none,some a.val)=s(v,w) := he
rcases Sym2.eq_iff.mp hh with hh|hh
· exact Or.inl ⟨hh.1.symm,hh.2.symm⟩
· exact Or.inr ⟨hh.1.symm,hh.2.symm⟩
theorem withSource_transfer_from_geometry (hlocal : G.LocallyFinite)
(A Input T : Set V) (hA : A.Finite) (q : (E⊕A) → unitInterval)
(π : V → ℤ) (H : ℤ) (J : Finset ℤ) (hJ : J.Nonempty)
(hH : ∀ j∈J, j<H) (hAlevel : ∀ a∈A, H ≤ π a)
(hInput : ∀ j∈J, ∀ v∈Input, π v<j)
(hcross : ∀ j∈J, ∀ e v w, G.ends e=s(v,w) → j ≤ π v → π w<j → π v=j)
(p : unitInterval) (hp : (p : ℝ)<1)
(hq : ∀ j∈J, ∀ e v w, G.ends e=s(v,w) → π v=j → π w<j → (q (Sum.inl e) : ℝ) ≤ p)
(D k M N : ℕ) (hD : ∀ j∈J, ∀ v, π v=j → (G.incidentEdges {v}).ncard ≤ D)
(hM : 0 < M) (hk : M*N ≤ k)
(C : V → V → Prop) (hrefl : ∀ v, C v v) (hsymm : ∀ {v w}, C v w → C w v)
(hfinite : ∀ v, {w | C v w}.Finite) (hbound : ∀ v, {w | C v w}.ncard ≤ M)
(F : ℤ → V → Set E) (B : ℤ → V → Set V)
(hF : ∀ j∈J, ∀ v, F j v ⊆ (G.inducedEdges {w | j ≤ π w})ᶜ)
(hconnect : ∀ j∈J, ∀ v, π v=j → ∀ b∈B j v, (G.openGraph (F j v)).Reachable v b)
(hdis : ∀ j∈J, ∀ v w, ¬C v w → Disjoint (F j v) (F j w))
(s₀ K η lam : ℝ) (hK : 0 ≤ K) (hη : 0 ≤ η) (hlam : 0 < lam)
(hs : ∀ j∈J, ∀ v, π v=j → s₀ ≤ (BernoulliProduct.law (fun e => q (Sum.inl e))).real (allOpen (F j v)))
(hN : K ≤ N*s₀)
(hmean : ∀ j∈J, ∀ v, π v=j →
1-η ≤ (BernoulliProduct.law (fun e => q (Sum.inl e))).real (G.seedHitEvent (B j v) T)) :
(BernoulliProduct.law q).real ((G.withSource A).hitEvent none (some '' Input)\
(G.withSource A).hitEvent none (some '' T)) ≤
1/((J.card : ℝ)*(1-(p : ℝ))^(D*k))+Real.exp (-K)+(K+1)*η/lam+lam := by
let π' : Option V → ℤ := Option.elim' H π
let C' : Option V → Option V → Prop := fun v w => match v,w with
| none,none => True
| some v,some w => C v w
| _,_ => False
let F' : ℤ → Option V → Set (E⊕A) := fun j => Option.elim' ∅ (fun v => Sum.inl '' F j v)
let B' : ℤ → Option V → Set (Option V) := fun j => Option.elim' ∅ (fun v => some '' B j v)
have hvA (j : ℤ) (hj : j∈J) (v : V) (hv : π v=j) : v∉A := by
intro hva
have := hAlevel v hva
have := hH j hj
omega
have hcross' (j : ℤ) (hj : j∈J) (e : E⊕A) (v w : Option V)
(he : (G.withSource A).ends e=s(v,w)) (hv : j ≤ π' v) (hw : π' w<j) : π' v=j := by
rcases G.withSource_edge_cases A e v w he with ⟨b,x,y,rfl,rfl,rfl,hb⟩|⟨a,rfl,hh⟩
· exact hcross j hj b x y hb hv hw
· obtain hh|hh := hh
· obtain ⟨rfl,rfl⟩ := hh
have := hAlevel a a.property
have := hH j hj
change π a<j at hw
omega
· obtain ⟨rfl,rfl⟩ := hh
change H<j at hw
have := hH j hj
omega
have hs' (j : ℤ) (hj : j∈J) (v : V) (hv : π v=j) :
s₀ ≤ (BernoulliProduct.law q).real (allOpen (Sum.inl '' F j v)) := by
have hm : Measurable (fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω : Set (E⊕A) → Set E) :=
Measurable.of_eval (fun edge => measurable_set_mem (Sum.inl edge))
have lawproj := BernoulliProduct.law_map_preimage q Sum.inl Sum.inl_injective
have heq : allOpen (Sum.inl '' F j v)=(fun ω : Set (E⊕A) => Sum.inl ⁻¹' ω) ⁻¹' allOpen (F j v) := by
ext ω
simp only [allOpen,mem_ofPred_eq,mem_preimage]
exact Set.image_subset_iff
rw [heq,measureReal_def,← Measure.map_apply hm ((blockSigma_le _) _ (measurable_allOpen_block (F j v))),lawproj]
exact hs j hj v hv
apply (G.withSource A).fresh_collar_transfer_from_geometry (G.withSource_local hlocal hA)
q π' none (some '' Input) (some '' T) J hJ (fun j hj => (hH j hj).le)
(by rintro j hj _ ⟨v,hv,rfl⟩; exact hInput j hj v hv) hcross' p hp
(by
intro j hj e v w he hv hw
rcases G.withSource_edge_cases A e v w he with ⟨b,x,y,rfl,rfl,rfl,hb⟩|⟨a,rfl,hh⟩
· exact hq j hj b x y hb hv hw
· obtain hh|hh := hh
· obtain ⟨rfl,rfl⟩ := hh
change H=j at hv
have := hH j hj
omega
· obtain ⟨rfl,rfl⟩ := hh
change H<j at hw
have := hH j hj
omega)
D k M N
(by
intro j hj v hv
cases v with
| none => have := hH j hj; change H=j at hv; omega
| some v =>
rw [G.withSource_incident A (hvA j hj v hv),Set.ncard_image_of_injective _ Sum.inl_injective]
exact hD j hj v hv)
hM hk C'
(by intro v; cases v; trivial; exact hrefl _)
(by intro v w h; cases v <;> cases w <;> simp_all only [C'])
(by
intro v
cases v with
| none =>
have he : {w | C' none w}={none} := by ext w; cases w <;> simp [C']
rw [he]
exact Set.finite_singleton _
| some v =>
have he : {w | C' (some v) w}=some '' {w | C v w} := by ext w; cases w <;> simp [C']
rw [he]
exact (hfinite v).image _)
(by
intro v
cases v with
| none =>
have he : {w | C' none w}={none} := by ext w; cases w <;> simp [C']
rw [he,Set.ncard_singleton]
omega
| some v =>
have he : {w | C' (some v) w}=some '' {w | C v w} := by ext w; cases w <;> simp [C']
rw [he,Set.ncard_image_of_injective _ (Option.some_injective V)]
exact hbound v)
F' B'
(by
intro j hj v
cases v with
| none => exact Set.empty_subset _
| some v =>
rintro e ⟨b,hb,rfl⟩ he
apply hF j hj v hb
intro z hz
exact he (some z) (by simpa [withSource] using hz))
(by
intro j hj v hv b hb
cases v with
| none => have := hH j hj; change H=j at hv; omega
| some v =>
obtain ⟨z,hz,rfl⟩ := hb
exact G.withSource_trial_reachable A (F j v) (hconnect j hj v hv z hz))
(by
intro j hj v w h
cases v <;> cases w
· simp [F']
· simp [F']
· simp [F']
· exact Set.disjoint_image_of_injective Sum.inl_injective (hdis j hj _ _ h))
s₀ K η lam hK hη hlam
(by
intro j hj v hv
cases v with
| none => have := hH j hj; change H=j at hv; omega
| some v => exact hs' j hj v hv)
hN
(by
intro j hj v hv
cases v with
| none => have := hH j hj; change H=j at hv; omega
| some v => exact (hmean j hj v hv).trans (G.withSource_seedHit A (B j v) T q))
end CriticalPercolation.BondGraph
end
end
end
end OAI
|